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1.
In this paper of the series, elliptic expansions in terms of the sectorial variables j (i) introduced recently in Paper IV (Sharaf, 1982) to regularize highly oscillating perturbations force of some orbital systems will be established analytically and computationally for the fifth and sixth categories. For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations; numerical results are included to provide test examples for constructing computational algorithms.  相似文献   

2.
In this paper of the series, elliptic expansions in terms of the sectorial variables j j introduced recently in Paper IV (Sharaf, 1982) to regularise highly oscillating perturbations force of some orbital systems will be established analytically and computationally for the seventh and eighth categories. For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations, numerical results are included to provide test examples for constructing computational algorithms.  相似文献   

3.
In this paper of the series, elliptic expansions in terms of the sectorial variables j (i) introduced by the author in Paper IV (Sharaf, 1982) to regularize the highly oscillating perturbation force of some orbital systems will be established analytically and computationally for the ninth, tenth, eleventh, and twelfth categories according to our adopted scheme of presentation drawn up in Paper V (Sharaf, 1983). For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computation, and numerical results are included to provide test examples for constructing computational algorithms. Finally, the first collection of completed elliptic expansions in terms of j (i) so explored will be given in Appendix A for the guidance of the reader.  相似文献   

4.
In this paper of the series, we arrive at the end of the second step of our regularization approach, and in which, elliptic expansions in terms of the sectorial variables j (i) introduced by the author in Paper IV (Sharaf, 1982b) to regularize the highly oscillating perturbation force of some orbital systems will be established analytically and computationally for the thirteenth, fourteenth, fifteenth, and sixteenth categories according to our adopted scheme of presentation drawn up in Paper V (Sharaf, 1983). For each of the elliptic expansions belonging to a category, literal analytic expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations, and numerical results are included to provide test examples for constructing computational algorithms. Finally, the second and the last collection of completed elliptic expansion will be given in Appendix B, such that, the materials of Appendix A of Paper VIII (Sharaf, 1985b) and those of Appendix B of the present paper provide the reader with the elliptic expansions in terms of j (i) so explored for the second step of our regularization approach.  相似文献   

5.
New expansions of elliptic motion based on considering the eccentricitye as the modulusk of elliptic functions and introducing the new anomalyw (a sort of elliptic anomaly) defined byw=u/2K–/2,g=amu–/2 (g being the eccentric anomaly) are compared with the classic (e, M), (e, v) and (e, g) expansions in multiples of mean, true and eccentric anomalies, respectively. These (q,w) expansions turn out to be in general more compact than the classical ones. The coefficients of the (e,v) and (e,g) expansions are expressed as the hypergeometric series, which may be reduced to the hypergeometric polynomials. The coefficients of the (q,w) expansions may be presented in closed (rational function) form with respect toq, k, k=(1–k 2)1/2,K andE, q being the Jacobi nome relatedk whileK andE are the complete elliptic integrals of the first and second kind respectively. Recurrence relations to compute these coefficients have been derived.on leave from Institute of Applied Astronomy, St.-Petersburg 197042, Russia  相似文献   

6.
Expansions of the functions (r/a)cos jv and (r/a)m sin jv of the elliptic motion are extended to highly eccentric orbits, 0.6627 ... <e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.  相似文献   

7.
A two-parameter time transformationdt=r 3/2(α0+α1 r)?1/2 dτ is proposed, where τ is the radial distance while α0 and α1 are, if not constants, at least conservative functions of positions and velocities. In Keplerian systems, the quadrature implied by the transformation may by carried out by elliptic functions. When α0=0, τ is the eccentric anomaly; if α1=0, then τ is the intermediate or elliptic anomaly. Considering several values of α0 and α1, numerical examples of the relation of thegeneralized elliptic anomaly τ with the classical and elliptic anomalies are given. Application of this transformation to some perturbed Kepler problems is briefly outlined.  相似文献   

8.
In this paper of the series, the third step of the author's regularization approach will be started by establishing the expansions of the functionX n (r) (, ,u) in terms of the sectorial variables j (i) introduced in Paper IV (Sharaf, 1982) to regularize the highly-oscillating perturbation force of some orbital systems. The literal analytical expressions for the Fourier expansion of the function will be explored in terms of j (i) for anyn positive integer,r any real number whatever the types and the number of sectors forming the divisions situation of the elliptic orbits may be. The basic computational materials of the theory will also be given and for which the method of solution, the recurrence formulae, and the general computational sequence for the coefficients are considered.  相似文献   

9.
In the first part of these notes new expressions—simpler than any previously obtained—are presented in integral form for the derivatives of the α n 0 -functions (required for an interpretation of the observed light changes of eclipsing variables) with respect to the fractional radiir 1, 2 and projected separation δ of their centres in terms of the modified Bessel functionsK 0, 1 (x) of the second kind; and utilized for establishing new asymptotic formulae for the computation of ‘boundary integrals’ of the formJ ?1 0 ,n(μ). In the second part of this paper, by a resort to bi-polar coordinates, we shall establish a new type of expansions for the α n 0 -functions valid for any type of eclipses, and converging faster than the expansions of the cross-correlation integral of the form (1) for α n 0 that have so far been established.  相似文献   

10.
In this paper of the series, literal analytical expressions for the coefficients of the Fourier series representation ofF will be established for anyx i ; withn, N positive integers 1 and | i | fori=1, 2,...n. Moreover, the recurrence formulae satisfied by these coefficients will also be established. Illustrative analytical examples and a full recursive computational algorithm, with its numerical results, are included. The applications of the recurrence formulae are also illustrated by their stencils. As a by-product of the analyses is an integral which we may call a complete elliptic integral of thenth kind, in which the known complete elliptic integrals (1st, 2nd and 3rd kinds) are special cases of it.  相似文献   

11.
The aim of the present paper is to establish two further series expansions (alternative to those given in Demircan, 1979a, b, c), for the observed light changes of eclipsing binary system. The coefficients of these expansions have also been expressed in the form of general series expansions in terms of the eclipse elementsr 1.2,i andL 1 of the spherical model on which all other distorted models may be based (Kopal, 1975, 1976) in an analysis in the frequency-domain.  相似文献   

12.
In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.  相似文献   

13.
The planar restricted 3-body problem, linearized in the neighborhood of Lagrangian equilibriaL 4 andL 5, has in general two distinct eigenvalues and their opposites. When they are pure imaginary and not multiples of each other, they generate two families of periodic solutions called long and short periodic families. This is essentially a consequence of the famous theorem of Liapunov (Siegel, 1956). We showed (Roels, 1971b) how to solve the problem when the eigenvalues are multiples of each other in building series with negative exponents instead of the integer expansions of Siegel (Roels and Lauterman, 1970). When the eigenvalues are equal, which is the case for the mass ratio of Routh, the problem was solved by Deprit and Henrard (1968) using formal series in ordinary unnormalized variables. That leads to very complicated series because of the use of variables that are not well adapted to the problem. The convergence of the series was proven by Meyer and Schmidt (1971). In this paper we solve the problem by using normalized variables. This brings us to build expansions with fractional exponents. So in summary, normalized variables generate integer series in the non-resonant cases, series with negative exponents in the case of resonancek≥3, and series with fractional exponents when the resonance is 1.  相似文献   

14.
In this paper we derive some recurrence formulae which can be used to calculate the Fourier expansions of the functions (r/a) n cosmv and (r/a) n sinmv in terms of the eccentric anomalyE or the mean anomalyM. We also establish a recurrence process for computing the series expansions for alln andm when the expansions of two basic series are known. These basic series were given in explicit form in the classical literature. The recurrence formulae are linear in the functions involved and thus make very simple the computation of the series.This work was supported by NASA contract No. NASr 54(06).—The paper was presented at the AIAA/AAS meeting, Princeton University, August 1969.  相似文献   

15.
Fourier expansions of elliptic motion functions in multiples of the true, eccentric, elliptic and mean anomalies are computed numerically by means of the fast Fourier transform. Both Hansen-like coefficients and their derivatives with respect to eccentricity of the orbit are considered. General behavior of the coefficients and the efficiency (compactness) of the expansions are investigated for various values of eccentricity of the orbit. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

16.
Hansen’s coefficients in the theory of elliptic motion with eccentricity e are studied as functions of the parameter η = (1 − e 2)1/2. Their analytic behavior in the complex η plane is described and some symmetry relations are derived. In particular, for every Hansen coefficient, multiplication by suitable powers of e and η results in an entire analytic function of η. Consequently, Hansen’s coefficients can be in principle computed by means of rapidly convergent series in powers of η. A representation of Hansen’s coefficients in terms of two entire functions of e 2 follows.   相似文献   

17.
The Fourier techniques developed so far for an analysis of eclipsing binary light curves have been re-discussed. The Fourier coefficients for the analysis have been derived in a simple form of series expansions, in terms of eclipse elements, valid for any type of eclipse (regardless of whetherr 1r 2).These coefficients may be utilized to solve the eclipse elements in terms of the observed characteristics of the light curves. A general relation between the observed quantitiesl and , and the eclipse elementsr 1,2,i andL 1 has also been given in the form of series expansions which can be used for the synthesis of the light curves.  相似文献   

18.
Following our recently published measurements of the rate coefficients for mutual neutralization, α, of the ionospherically important reactions NO+ + NO2?(α1) and NO+ + NO3?(α2) carried out in ion-ion flowing afterglow plasmas at 300 K, we have determined the mutual neutralization rates for the water cluster ion H3O+ · (H2O)3 with a mixture of several negative ions which are known to exist in the D region. The α coefficients for these cluster ion reactions do not differ significantly from alpha;1 and α2, all of these reactions having α ?6 × 10?8 cm3/sec which is significantly smaller than values usually adopted in ionospheric calculations. Current information on the ionic composition of the D region and the implications of the present results to de-ionization rate calculations are discussed.  相似文献   

19.
In this paper of the series, the time transform and the explicit exact forms of the time will be established in terms of the sectorial variables j (i) introduced in Paper IV (Sharaf, 1982) to regularize the highly oscillating perturbation force. Simple recurrence formulae are given to facilitate the computations. The formulations are general in the sense that they are valid whatever the types and the number of sectors forming the divisions situation of the elliptic orbit may be. Moreover, the constants of integration for the explicit forms of the time are determined in a way that it gives for these forms its generality during any revolution of the body in its Keplerian orbit.  相似文献   

20.
We performed a series of CCD observations in BVRI bands of a celestial region in cluster M67 (NGC 2682) to study the photometric system of the Kyiv network telescope. The signal recording system consists of a CGE-1400 telescope, Celestron #94175 focal reducer, an automatic turret with a set of UBVRI filters, and a Rolera Mgi CCD camera. The operating field of the system is 10.62′ × 10.62′. CCD images are processed in the MIDAS/ROMAFOT package. The reduction coefficients of the instrumental photometric system relative to Johnson’s BVRI system are determined. The resulted value of the module of distance to cluster M67 V ? M V = 9.63 m does not contradict the results of other researchers. The mean square errors for one determination of stellar magnitude for different bands are 0.02–0.12 m . The errors in determining equatorial coordinates in the TYCHO-2 catalog system are ± 1″.  相似文献   

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